Far less has been written about this issue for semiclassical approximations of quantum dynamics based on Glauber coherent states [1]} or Gaussian wave packets [2]}, [3]}, although they appear in an extremely wide variety of physical contexts [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}. In the former, one is making a semiclassical approximation to a coherent state representation of a path integral [11]}, [12]}, and in the latter, one is carrying out the so-called generalized Gaussian wave packet dynamics (GGWPD) approximation, which is proven equivalent to a complex version of time-dependent WKB theory [3]}. The underlying mathematics is essentially identical for these two very different physical contexts. Both necessitate a saddle point approximation instead of a stationary phase one and require complex trajectories. For a typical saddle trajectory the determinant, being complex, never vanishes. Instead, the phase just accumulates, and for the most part the correct phase for the prefactor is the inverse of half the determinant's total accumulated phase [3]}, [12]}. As such, there appears to be no need to discuss a Maslov index. Nevertheless, it is helpful to translate the phase calculations into a Maslov index approach upon which later sections of this paper rely. By calculating the determinant's total accumulated phase angle modulo \(\pi \) , its square root is brought into the first quadrant, call that phase \(\theta \) . The integer number of \(\pi \)  being dropped in the process becomes the index \(\nu \) , which gives a multiplying phase of \(\exp (-i\nu \pi /2)\)  for the prefactor multiplied by \(\exp (-i\theta )\) .
